| L(s) = 1 | − 2-s + 1.08·3-s + 4-s − 3.61·5-s − 1.08·6-s − 1.95·7-s − 8-s − 1.81·9-s + 3.61·10-s + 1.46·11-s + 1.08·12-s + 4.87·13-s + 1.95·14-s − 3.92·15-s + 16-s + 0.680·17-s + 1.81·18-s + 4.42·19-s − 3.61·20-s − 2.12·21-s − 1.46·22-s − 2.12·23-s − 1.08·24-s + 8.03·25-s − 4.87·26-s − 5.23·27-s − 1.95·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.627·3-s + 0.5·4-s − 1.61·5-s − 0.443·6-s − 0.739·7-s − 0.353·8-s − 0.606·9-s + 1.14·10-s + 0.442·11-s + 0.313·12-s + 1.35·13-s + 0.523·14-s − 1.01·15-s + 0.250·16-s + 0.165·17-s + 0.428·18-s + 1.01·19-s − 0.807·20-s − 0.464·21-s − 0.312·22-s − 0.444·23-s − 0.221·24-s + 1.60·25-s − 0.955·26-s − 1.00·27-s − 0.369·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7737849176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7737849176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 17 | \( 1 - 0.680T + 17T^{2} \) |
| 19 | \( 1 - 4.42T + 19T^{2} \) |
| 23 | \( 1 + 2.12T + 23T^{2} \) |
| 29 | \( 1 + 9.83T + 29T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 1.13T + 47T^{2} \) |
| 53 | \( 1 + 0.265T + 53T^{2} \) |
| 59 | \( 1 - 4.04T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 + 1.20T + 71T^{2} \) |
| 73 | \( 1 - 6.93T + 73T^{2} \) |
| 79 | \( 1 + 6.22T + 79T^{2} \) |
| 83 | \( 1 + 0.628T + 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.075732602701501006579285269940, −7.61703840393651630195022171368, −6.93145110432114907674765093914, −6.12651221380727340607308692277, −5.32683017256201223090694820180, −3.93612355170610662723267555767, −3.56049136361699291836092009987, −3.04422975178983562116628348065, −1.72283977394301199966776226338, −0.49348822548718153076937545791,
0.49348822548718153076937545791, 1.72283977394301199966776226338, 3.04422975178983562116628348065, 3.56049136361699291836092009987, 3.93612355170610662723267555767, 5.32683017256201223090694820180, 6.12651221380727340607308692277, 6.93145110432114907674765093914, 7.61703840393651630195022171368, 8.075732602701501006579285269940
